Similar analysis can be performed in the case \(k=4\) . For example, one can use the procedure IntegralQuarticPoints of the Magma computational package ([1]}) and find that all the integral points on \(C_{4}\)  are:
\((1,\pm 3), (0, \pm 1), (7, \pm 141), (-2, \pm 15), (-3, \pm 29), (172, \pm 83679).\) 
